**Note**: This question was posted in error, and should be closed as no longer relevant. The correct question is posted at Help with this system of Diophantine equations **(End of note)**

For a research problem that I'm working on, I need to solve this Diophantine equation:-

$a^3+b^3+c^3-3d=-83449$, where $a,b,c,d>0$ are all DISTINCT positive integers and$ a,b,c∉ ${$2,9,15,16,33,34$}.

How does one go about solving this? Is brute-force the only possible way? Or could there be a case that no integer solutions exist for this equation?

Also, are there any online computing engines, that allow me to set constraints, and solve Diophantine equations of this sort?

Any and all help is appreciated! Thanks!